L(s) = 1 | + (−0.793 + 0.608i)5-s + (−0.258 + 0.965i)7-s + (0.991 − 0.130i)11-s + (−0.130 + 0.991i)13-s − i·17-s + (−0.923 + 0.382i)19-s + (0.258 + 0.965i)23-s + (0.258 − 0.965i)25-s + (−0.608 + 0.793i)29-s + (0.5 − 0.866i)31-s + (−0.382 − 0.923i)35-s + (−0.923 − 0.382i)37-s + (0.258 + 0.965i)41-s + (−0.991 + 0.130i)43-s + (−0.866 + 0.5i)47-s + ⋯ |
L(s) = 1 | + (−0.793 + 0.608i)5-s + (−0.258 + 0.965i)7-s + (0.991 − 0.130i)11-s + (−0.130 + 0.991i)13-s − i·17-s + (−0.923 + 0.382i)19-s + (0.258 + 0.965i)23-s + (0.258 − 0.965i)25-s + (−0.608 + 0.793i)29-s + (0.5 − 0.866i)31-s + (−0.382 − 0.923i)35-s + (−0.923 − 0.382i)37-s + (0.258 + 0.965i)41-s + (−0.991 + 0.130i)43-s + (−0.866 + 0.5i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2121081971 + 0.6857404448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2121081971 + 0.6857404448i\) |
\(L(1)\) |
\(\approx\) |
\(0.7511285643 + 0.2963960791i\) |
\(L(1)\) |
\(\approx\) |
\(0.7511285643 + 0.2963960791i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.793 + 0.608i)T \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
| 11 | \( 1 + (0.991 - 0.130i)T \) |
| 13 | \( 1 + (-0.130 + 0.991i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (-0.923 + 0.382i)T \) |
| 23 | \( 1 + (0.258 + 0.965i)T \) |
| 29 | \( 1 + (-0.608 + 0.793i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.923 - 0.382i)T \) |
| 41 | \( 1 + (0.258 + 0.965i)T \) |
| 43 | \( 1 + (-0.991 + 0.130i)T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.382 + 0.923i)T \) |
| 59 | \( 1 + (-0.793 + 0.608i)T \) |
| 61 | \( 1 + (0.608 - 0.793i)T \) |
| 67 | \( 1 + (-0.991 - 0.130i)T \) |
| 71 | \( 1 + (-0.707 - 0.707i)T \) |
| 73 | \( 1 + (-0.707 + 0.707i)T \) |
| 79 | \( 1 + (0.866 - 0.5i)T \) |
| 83 | \( 1 + (0.793 + 0.608i)T \) |
| 89 | \( 1 + (0.707 + 0.707i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.91171956605024115826496352813, −22.29498619594166502649075751455, −21.019759272449661672127488852978, −20.30496663729117059845365047055, −19.54648199262769241503011432487, −19.11408844658959061733357701510, −17.52891609539879359119363875240, −17.074973173692075850835769816739, −16.275662264178104394090109540454, −15.24492870522037262322920266905, −14.6051422457370218616533702400, −13.35463392123656067806100403176, −12.70638485161448603515216104215, −11.86988632672110811252907490306, −10.79509858189016098264740554264, −10.10353430558634480353541186477, −8.82225420243921330098843879787, −8.174353744166908044995599518796, −7.12750282050454008151853321048, −6.29983435037605121801903581663, −4.90173513799336292202411942013, −4.07708873643580332904573341803, −3.3109514956429265247979550812, −1.610647164720834223296935029929, −0.373531197811871191807233132421,
1.6940418720738943954020293219, 2.89299716678211298402810325077, 3.80651979693962518173527115131, 4.84538203236722268133929541990, 6.17561980112379346426795678528, 6.837432438240827996329842185562, 7.86555905160919550767710245746, 8.97648756445796791714367629748, 9.55608155595098045721317269291, 10.9313247672585304547750680809, 11.72024160114830338842729395306, 12.16460652566559627966597173939, 13.43914699735093505332919438509, 14.51777639250810306763361362402, 15.01726970079729916750509755155, 16.00441431240366075007346238262, 16.69255940638057710319422213868, 17.84748394771146151434467989636, 18.86668491586205123119460840723, 19.14247193432188167298331349405, 20.06962311461794811413982905249, 21.25936049065357369421425924183, 21.97076054914318258873416574593, 22.68748075208053400708446902303, 23.45341014313376052186931252624